TITLE: Program for computing group cohomology
QUESTION [15 upvotes]: Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space? 
I mainly care about infinite groups.

REPLY [18 votes]: GAP can be used to compute free $\mathbb ZG$-resolutions of $\mathbb Z$ for some infinite groups G. These resolutions can then be used to compute homology and cohomology of $G$. I hope the following examples give a flavour of the kind of infinite groups that can be handled at present. (A range of finite groups can also be handled.)

EXAMPLE   $H_{99}(SL_2(\mathbb Z[1/7]),\mathbb Z) = \mathbb Z_4 \oplus \mathbb Z_{12}$
gap> R:=ResolutionSL2Z(7,100);;
gap> Homology(TensorWithIntegers(R),99);
[ 4, 12 ]

EXAMPLE $H_3(SL_3(\mathbb Z),\mathbb Z) = \mathbb Z_{12} \oplus \mathbb Z_{12}$ 
gap> R:=ResolutionArithmeticGroup("SL(3,Z)",4);;
gap> Homology(TensorWithIntegers(R),3);
[ 12, 12 ]

EXAMPLE $H_3(SL_2(O_{-11}),\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_{24}$ where $O_{-11}$ is the ring of integers of $\mathbb Q(\sqrt{-11})$. 
gap> R:=ResolutionArithmeticGroup("SL(2,O-11)",4);;
gap> Homology(TensorWithIntegers(R),3);
[ 2, 24 ]

EXAMPLE $H_3(G,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z$ for $G$ the space group P62. 
gap> G:=Image(IsomorphismPcpGroup(SpaceGroupBBNWZ("P62")));;
gap> R:=ResolutionAlmostCrystalGroup(G,4);;
gap> Homology(TensorWithIntegers(R),3);
[ 2, 2, 0 ]

EXAMPLE  $H_3(G,\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z^{110}$ for $G$ the Heisenberg group of degree 5.
gap> R:=ResolutionNilpotentGroup(HeisenbergPcpGroup(5),4);;
gap> Homology(TensorWithIntegers(R),3);
[ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

EXAMPLE $H_3(G,\mathbb Z) = \mathbb Z_2$ for $G$ the braid group on eight braids (or seven generators). 
gap> Dynkin:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];;
gap> R:=ResolutionArtinGroup(Dynkin,4);;
gap> Homology(TensorWithIntegers(R),3);
[ 2 ]